We focus on the problem of finding a non-linear classification function that lies in a Reproducing Kernel Hilbert Space (RKHS) both from the primal point of view (finding a perfect separator when one exists) and the dual point of view (giving a certificate of non-existence), with special focus on generalizations of two classical schemes - the Perceptron (primal) and Von-Neumann (dual) algorithms. We cast our problem as one of maximizing the regularized normalized hard-margin (ρ) in an RKHS and use the Representer Theorem to rephrase it in terms of a Mahalanobis dot-product/semi-norm associated with the kernel’s (normalized and signed) Gram matrix. We derive an accelerated smoothed algorithm with a convergence rate of \tfrac\sqrt \log nρ given n separable points, which is strikingly similar to the classical kernelized Perceptron algorithm whose rate is \tfrac1ρ^2. When no such classifier exists, we prove a version of Gordan’s separation theorem for RKHSs, and give a reinterpretation of negative margins. This allows us to give guarantees for a primal-dual algorithm that halts in \min{\tfrac\sqrt n|ρ|, \tfrac\sqrt nε} iterations with a perfect separator in the RKHS if the primal is feasible or a dual ε-certificate of near-infeasibility.

This paper considers a wide spectrum of regularized stochastic optimization problems where both the loss function and regularizer can be non-smooth. We develop a novel algorithm based on the regularized dual averaging (RDA) method, that can simultaneously achieve the optimal convergence rates for both convex and strongly convex loss. In particular, for strongly convex loss, it achieves the optimal rate of $O(\frac{1}{N}+\frac{1}{N^2})$ for $N$ iterations, which improves the best known rate $O(\frac{\log N }{N})$ of previous stochastic dual averaging algorithms. In addition, our method constructs the final solution directly from the proximal mapping instead of averaging of all previous iterates. For widely used sparsity-inducing regularizers (e.g., $\ell_1$-norm), it has the advantage of encouraging sparser solutions. We further develop a multi-stage extension using the proposed algorithm as a subroutine, which achieves the uniformly-optimal rate $O(\frac{1}{N}+\exp\{-N\})$ for strongly convex loss.